Most likely it was simply physics; it grew to a sufficient size that the physical forces on it were sufficient to tear the cell into two parts. That's what lipid vesicles do.
Of course it was then just luck whether the two halves contained what they needed to survive. Sometimes one part would contain everything; sometimes both of them wouldn't. This, then, applies a strong selective pressure towards mechanisms that control the process of division and ensure that both daughter cells contain the correct elements - and we have the beginnings of mitosis.
I believe that's a pretty accurate description of the hypothesis presented, yes. Although I'd note that it's just one suggestion; one that is popular among those on this board, but by no means the "accepted" account. An alternative view has life evolving initially in the microstructure of rocks, and only later developing cell membranes.
I feel the need to ask if a life-form has been developed in a lab? or, is the current state of development still chemical compounds?
No origin of life experiment so far has come close to anything that might be called living from non-replicating components.